I mostly have two points to raise, one that doesn't seem to have been mentioned previously, the other more an extended comment on another answer.
My first point is that (at least with the bodies I've used) the full-frame and the crop-frame camera had almost the same pixel density on the sensor. That let me shoot at about the same, or just slightly longer focal length on the full-frame body while retaining about the same level of detail (or somewhat more) but getting a lot looser framing, so in fast-moving sports it was much easier to be sure of getting all the action in the frame. Then I could crop out the extra later. Sometimes I cropped about even all around, so it was essentially the same as if I'd been shooting with a cropped sensor.
Other times, however, I picked an off-center crop. In those cases, it's a pretty fair bet that if I'd been shooting with a cropped sensor, I just wouldn't have gotten the shot.
As far as speed of shooting (frame rate) goes, however, I have to disagree with @Fake Name. At first glance, frame rate seems like it would be important. I think a lot of beginners are "tricked" into spending more for cameras with high frame rates based (in large part) on "pro" cameras that also have higher frame rates.
At least in my experience, however, if you know what you're doing, frame rate is about 99% irrelevant. Consider an example:
This was at a professional baseball game, and that was a fastball moving close to 100 MPH. 100 MPH works out to ~147 feet per second, so even at 10 frames per second, your consecutive shots will have the ball nearly 15 feet apart. If you depend on a fast frame rate, you probably have to shoot practically every pitch for at least one entire game (and probably two or three) before you can depend on getting a shot like that.
That, however, was shot with a Konica-Minolta D7D, which had a maximum frame rate of about 3 FPS if memory serves -- and that was still completely irrelevant, because I had it in single-shot mode anyway. Despite (or really, because of) that I was able to capture pictures on this order about one out of three (or so) pitches where the batter actually swung so there was at least some chance.
Oh, as far as detail goes, that started as a 6 MP shot, and was then down-sized quite a bit. Realistically, its biggest shortcoming is that this was a night-game, and the D7D did not handle high ISOs well at all (so I was shooting at ISO 100).
I should add that this should not be read as my bragging about my incredible reaction speed or anything like that either -- quite the contrary, I'm pretty sure that most people can probably do about as well (at least with some practice) and there are undoubtedly at least a few who can do quite a bit better -- for that matter, I'm pretty sure I'd do better if I did it more often.